Intersecting integer partitions: star bounds and counterexamples at every scale
Abstract
Two integer partitions t-intersect if they have at least t common parts, counted with multiplicity. We study the largest t-intersecting families of integer partitions of n into exactly k positive parts. The canonical t-star consists of the partitions containing at least t ones. Applying Kupavskii's weak-spread theorem, we prove that this star is largest whenever n≥ Ak3, for every fixed A>24 and all sufficiently large k, uniformly over 1≤ t<k. We also give counterexamples to Borg's conjecture at every intersection scale: for all sufficiently large k and for every 1≤ t≤ ⌈ k/4⌉, we describe t-intersecting families on integer partitions of n=⌊ tk2/3⌋ into k parts which are strictly larger than the corresponding canonical stars. Moreover we present a family of counterexamples to Borg's conjectures about largest t-intersecting families of integer partitions of n without restricting the number of parts.
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BibTeX
@misc{person2026intersecting,
title = {{Intersecting integer partitions: star bounds and counterexamples at every scale}},
author = {Yury Person and Thomas Schweser},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01747},
eprint = {2610.01747},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01747},
}
RIS
TY - UNPB
TI - Intersecting integer partitions: star bounds and counterexamples at every scale
AU - Yury Person
AU - Thomas Schweser
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.01747
UR - https://arxiv.org/abs/2610.01747
AB - Two integer partitions t-intersect if they have at least t common parts, counted with multiplicity. We study the largest t-intersecting families of integer partitions of n into exactly k positive parts. The canonical t-star consists of the partitions containing at least t ones. Applying Kupavskii's weak-spread theorem, we prove that this star is largest whenever n≥ Ak^3, for every fixed A>24 and all sufficiently large k, uniformly over 1≤ t<k. We also give counterexamples to Borg's conjecture at every intersection scale: for all sufficiently large k and for every 1≤ t≤ ⌈ k/4⌉, we describe t-intersecting families on integer partitions of n=⌊ tk^2/3⌋ into k parts which are strictly larger than the corresponding canonical stars. Moreover we present a family of counterexamples to Borg's conjectures about largest t-intersecting families of integer partitions of n without restricting the number of parts.
ER -